Derivative of 1+27x-2xsqrt(x)/(x)

1. Differentiate $\left(27 x + 1\right) - \frac{\sqrt{x} 2 x}{x}$ term by term:

   1. Differentiate $27 x + 1$ term by term:

      1. The derivative of the constant $1$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $27$

      The result is: $27$

   2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = x^{\frac{3}{2}}$ and $g{\left(x \right)} = x$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

            1. Apply the power rule: $x^{\frac{3}{2}}$ goes to $\frac{3 \sqrt{x}}{2}$

            To find $\frac{d}{d x} g{\left(x \right)}$:

            1. Apply the power rule: $x$ goes to $1$

            Now plug in to the quotient rule:

            $\frac{1}{2 \sqrt{x}}$

         So, the result is: $\frac{1}{\sqrt{x}}$

      So, the result is: $- \frac{1}{\sqrt{x}}$

   The result is: $27 - \frac{1}{\sqrt{x}}$

The answer is:

$$27 - \frac{1}{\sqrt{x}}$$